$r$-Critical Numbers of Natural Intervals

The critical number $cr(r,n)$ of natural intervals $[r,n]$ was introduced by Herzog, Kaplan and Lev in 2014. The critical number $cr(r,n)$ is the smallest integer $t$ satisfying the following conditions: (i) every sequence of integers $S=\{r_1=r\leq r_2\leq \dotsb\leq r_k\}$ with $r_1+r_2 +\dotsb +r_k=n$ and $k\geq t$ has the following property: every integer between $r$ and $n-r$ can be written as a sum of distinct elements of $S$, and (ii) there exists $S$ with $k=t$, which satisfies that property. In this paper we study a variation of the critical number $cr(r,n)$ called the $r$-critical number $rcr(r,n)$. We determine the value of $rcr(r,n)$ for all $r,n$ satisfying $r\mid n$.